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%I #10 Oct 01 2017 00:07:45
%S 60,180,1050,5040,29244,161340,1046250,4825800,27790266,145126548,
%T 843333015,4466836920,26967624184,137243187108,789854179074,
%U 4306147750200,24711052977222,134216193832908,797987818325009,4240082199867228
%N a(n) is the number of representative six-color bracelets (necklaces with turning over allowed; D_6 symmetry) with n beads, for n >= 6.
%C This is the sixth column (m = 6) of triangle A213940.
%C The relevant p(n,6)= A008284(n, 6) representative color multinomials have exponents (signatures) from the six-part partitions of n, written with nonincreasing parts. E.g., n = 8: [3,1,1,1,1,1] and [2,2,1,1,1,1] (p(8,6)=2). The corresponding representative bracelets have the six-color multinomials c[1]^3*c[2]*c[3]*c[4]*c[5]*c[6] and c[1]^2*c[2]^2*c[3]*c[4]*c[5]*c[6].
%C See A056361 for the numbers if also color permutations for D_6 inequivalent bracelets are allowed. (_Andrew Howroyd_ induced me to look at these bracelets.)
%F a(n) = A213940(n, 6), n >= 6.
%F a(n) = Sum_{k=b(n, 6)..b(n, 7)-1} A213939(n, k), for n >= 7, with b(n, m) = A214314(n, m) the position where the first m-part partition of n appears in the Abramowitz-Stegun ordering of partitions (see A036036 for the reference and a historical comment), and a(6) = A213939(6, b(6,6)) = A213939(6, 11) = 60.
%e a(6) = A213940(6,6) = A213939(6, 11) = 60 from the representative bracelets (with colors j for c(j), j=1..6) permutations of (1, 2, 3, 4, 5, 6) modulo D_6 (dihedral group) symmetry, i.e., modulo cyclic or anti-cyclic operations. E.g., (1, 2, 3, 4, 6, 5) == (2, 3, 4, 6, 5, 1) == (6, 4, 3, 2, 1, 5) == ..., but (1, 2, 3, 4, 6, 5) is not equivalent to (1, 2, 3, 4, 5, 6). If color permutation is also allowed, then there is only one possibility (see A056361(6) = 1).
%Y Cf. A008284, A036036, A056361, A213940, A214311, A214314.
%K nonn
%O 6,1
%A _Wolfdieter Lang_, Sep 30 2017
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